CAREER: NLApack: Software for Numerical Algebraic Geometry

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In this Career project, the investigator creates a softwareplatform (called NLApack) for Numerical Algebraic Geometry,trains students in the development and application ofmathematical software, and broadens the graduate education ofstudents in the areas of mathematics, computational science, andengineering. In a first stage, the algorithms for the numericalirreducible decomposition of the solution set of a polynomialsystem are refined and combined into a blackbox program. Besidessoftware, another outcome of this stage is the development of atextbook for a course introducing students to symbolic andscientific computing. The second stage of the project focuses onthree specific topics: pole placement, overconstrained linkages,and multi-body dynamics. A result of this stage is a collectionof software tools specialized for those application fields, casestudies, and benchmarks. Throughout, students are trained inmathematics and computing and in important applications areas. The principal investigator, his collaborators, and hisgraduate students are working in the areas of numerical analysisand computer algebra. While computer algebra solves mathematicalproblems in an exact symbolic fashion, as one would by hand,numerical analysis calculates with limited precision on inputdata often only approximately known. The software developed inthis project combines the symbolic and numeric approach. Mostscientists seek insight (not just numbers) in the form ofequations to reveal new relations between the importantparameters in their models. But these models contain approximatedata that cannot be handled directly by current symbolic methods.The investigator develops symbolic-numeric methods that willmake computer algebra more relevant to scientific computing, andtrains students who will adept at both numeric and symboliccomputation, with significant payoffs for disciplines that dependon large-scale scientific computing.